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%\usepackage[mathlines]{lineno}% Enable numbering of text and display math |
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\date{\today}% It is always \today, today, |
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% but any date may be explicitly specified |
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\begin{abstract} |
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This document includes useful relationships for computing the |
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interactions between fields and field gradients and point multipolar |
| 74 |
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representations of molecular electrostatics. We also provide |
| 75 |
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explanatory derivations of a number of relationships used in the |
| 76 |
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main text. This includes the Boltzmann averages of quadrupole |
| 77 |
+ |
orientations, and the interaction of a quadrupole with the |
| 78 |
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self-generated field gradient. This last relationship is assumed to |
| 79 |
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be zero in the main text but is explicitly shown to be zero here. |
| 80 |
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\end{abstract} |
| 81 |
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|
| 82 |
|
\maketitle |
| 83 |
|
|
| 84 |
< |
This document contains derivations of useful relationships for |
| 85 |
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electric field gradients and their interactions with point multipoles. |
| 86 |
< |
We rely heavily on both the notation and results from Torres del |
| 87 |
< |
Castillo and Mend\'{e}z Garido.\cite{Torres-del-Castillo:2006uo} In |
| 88 |
< |
this work, tensors are expressed in Cartesian components, using at |
| 89 |
< |
times a dyadic notation. This proves quite useful for our work as we |
| 90 |
< |
employ toroidal boundary conditions in our simulations, and these are |
| 91 |
< |
easily implemented in Cartesian coordinate systems. |
| 84 |
> |
\section{Generating Uniform Field Gradients} |
| 85 |
> |
One important task in performing out the simulations mentioned in the |
| 86 |
> |
main text was to generate uniform electric field gradients. We rely |
| 87 |
> |
heavily on both the notation and results from Torres del Castillo and |
| 88 |
> |
Mend\'{e}z Garido.\cite{Torres-del-Castillo:2006uo} In this work, |
| 89 |
> |
tensors were expressed in Cartesian components, using at times a |
| 90 |
> |
dyadic notation. This proves quite useful for computer simulations |
| 91 |
> |
that make use of toroidal boundary conditions. |
| 92 |
|
|
| 93 |
|
An alternative formalism uses the theory of angular momentum and |
| 94 |
|
spherical harmonics and is common in standard physics texts such as |
| 95 |
< |
Jackson,\cite{Jackson98} Morse and Feshbach, and Baym. Because this |
| 96 |
< |
approach has its own advantages, relationships are provided below |
| 97 |
< |
comparing that terminology to the Cartesian tensor notation. |
| 95 |
> |
Jackson,\cite{Jackson98} Morse and Feshbach,\cite{Morse:1946zr} and |
| 96 |
> |
Stone.\cite{Stone:1997ly} Because this approach has its own |
| 97 |
> |
advantages, relationships are provided below comparing that |
| 98 |
> |
terminology to the Cartesian tensor notation. |
| 99 |
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|
| 100 |
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The gradient of the electric field, |
| 101 |
|
\begin{equation*} |
| 117 |
|
electrostatic potential that generates a uniform gradient may be |
| 118 |
|
written: |
| 119 |
|
\begin{align} |
| 120 |
< |
\Phi(x, y, z) =\; -g_o & \left(\frac{1}{2}(a_1\;b_1 - \frac{cos\psi}{3})\;x^2+\frac{1}{2}(a_2\;b_2 - \frac{cos\psi}{3})\;y^2 + \frac{1}{2}(a_3\;b_3 - \frac{cos\psi}{3})\;z^2 \right. \\ |
| 121 |
< |
& \left. + \frac{(a_1\;b_2 + a_2\;b_1)}{2} x\;y + \frac{(a_1\;b_3 + a_3\;b_1)}{2} x\;z + \frac{(a_2\;b_3 + a_3\;b_2)}{2} y\;z \right) . |
| 120 |
> |
\Phi(x, y, z) =\; -\frac{g_o}{2} & \left(\left(a_1b_1 - |
| 121 |
> |
\frac{cos\psi}{3}\right)\;x^2+\left(a_2b_2 |
| 122 |
> |
- \frac{cos\psi}{3}\right)\;y^2 + |
| 123 |
> |
\left(a_3b_3 - |
| 124 |
> |
\frac{cos\psi}{3}\right)\;z^2 \right. \nonumber \\ |
| 125 |
> |
& + (a_1b_2 + a_2b_1)\; xy + (a_1b_3 + a_3b_1)\; xz + (a_2b_3 + a_3b_2)\; yz \bigg) . |
| 126 |
|
\label{eq:appliedPotential} |
| 127 |
|
\end{align} |
| 128 |
|
Note $\mathbf{a}\cdot\mathbf{a} = \mathbf{b} \cdot \mathbf{b} = 1$, |
| 129 |
|
$\mathbf{a} \cdot \mathbf{b}=\cos \psi$, and $g_0$ is the overall |
| 130 |
< |
strength of the potential. |
| 130 |
> |
strength of the potential. |
| 131 |
|
|
| 132 |
< |
An alternative to this notation is to write an electrostatic potential |
| 133 |
< |
that generates a uniform gradient using the notation of Morse and |
| 134 |
< |
Feshbach, |
| 132 |
> |
Taking the gradient of Eq. (\ref{eq:appliedPotential}), we find the |
| 133 |
> |
field due to this potential, |
| 134 |
> |
\begin{equation} |
| 135 |
> |
\mathbf{E} = -\nabla \Phi |
| 136 |
> |
=\frac{g_o}{2} \left(\begin{array}{ccc} |
| 137 |
> |
2(a_1 b_1 - \frac{cos\psi}{3})\; x & +\; (a_1 b_2 + a_2 b_1)\; y & +\; (a_1 b_3 + a_3 b_1)\; z \\ |
| 138 |
> |
(a_2 b_1 + a_1 b_2)\; x & +\; 2(a_2 b_2 - \frac{cos\psi}{3})\; y & +\; (a_2 b_3 + a_3 b_3)\; z \\ |
| 139 |
> |
(a_3 b_1 + a_3 b_2)\; x & +\; (a_3 b_2 + a_2 b_3)\; y & +\; 2(a_3 b_3 - \frac{cos\psi}{3})\; z |
| 140 |
> |
\end{array} \right), |
| 141 |
> |
\label{eq:CE} |
| 142 |
> |
\end{equation} |
| 143 |
> |
while the gradient of the electric field in this form, |
| 144 |
> |
\begin{equation} |
| 145 |
> |
\mathsf{G} = \nabla\mathbf{E} |
| 146 |
> |
= \frac{g_o}{2}\left(\begin{array}{ccc} |
| 147 |
> |
2(a_1\; b_1 - \frac{cos\psi}{3}) & (a_1\; b_2 \;+ a_2\; b_1) & (a_1\; b_3 \;+ a_3\; b_1) \\ |
| 148 |
> |
(a_2\; b_1 \;+ a_1\; b_2) & 2(a_2\; b_2 \;- \frac{cos\psi}{3}) & (a_2\; b_3 \;+ a_3\; b_3) \\ |
| 149 |
> |
(a_3\; b_1 \;+ a_3\; b_2) & (a_3\; b_2 \;+ a_2\; b_3) & 2(a_3\; b_3 \;- \frac{cos\psi}{3}) |
| 150 |
> |
\end{array} \right), |
| 151 |
> |
\label{eq:GC} |
| 152 |
> |
\end{equation} |
| 153 |
> |
is uniform over the entire space. Therefore, to describe a uniform |
| 154 |
> |
gradient in this notation, two unit vectors ($\mathbf{a}$ and |
| 155 |
> |
$\mathbf{b}$) as well as a potential strength, $g_0$, must be |
| 156 |
> |
specified. As expected, this requires five independent parameters. |
| 157 |
> |
|
| 158 |
> |
The common alternative to the Cartesian notation expresses the |
| 159 |
> |
electrostatic potential using the notation of Morse and |
| 160 |
> |
Feshbach,\cite{Morse:1946zr} |
| 161 |
|
\begin{equation} \label{eq:quad_phi} |
| 162 |
< |
\Phi(x,y,z) = - \left[ a_{20} \frac{2 z^2 -x^2 - y^2}{2} |
| 162 |
> |
\Phi(x,y,z) = -\left[ a_{20} \frac{2 z^2 -x^2 - y^2}{2} |
| 163 |
|
+ 3 a_{21}^e \,xz + 3 a_{21}^o \,yz |
| 164 |
< |
+ 6a_{22}^e \,xy + 3 a_{22}^o (x^2 - y^2) \right] . |
| 164 |
> |
+ 6a_{22}^e \,xy + 3 a_{22}^o (x^2 - y^2) \right]. |
| 165 |
|
\end{equation} |
| 166 |
|
Here we use the standard $(l,m)$ form for the $a_{lm}$ coefficients, |
| 167 |
|
with superscript $e$ and $o$ denoting even and odd, respectively. |
| 168 |
|
This form makes the functional analogy to ``d'' atomic states |
| 169 |
< |
apparent. The gradient of the electric field in this form is: |
| 169 |
> |
apparent. |
| 170 |
> |
|
| 171 |
> |
Applying the gradient operator to Eq. (\ref{eq:quad_phi}) the electric |
| 172 |
> |
field due to this potential, |
| 173 |
> |
\begin{equation} |
| 174 |
> |
\mathbf{E} = -\nabla \Phi |
| 175 |
> |
= \left(\begin{array}{ccc} |
| 176 |
> |
\left( 6a_{22}^o -a_{20} \right)\; x &+\; 6a_{22}^e\; y &+\; 3a_{21}^e\; z \\ |
| 177 |
> |
6a_{22}^e\; x & -\; (a_{20} + 6a_{22}^o)\; y & +\; 3a_{21}^o\; z \\ |
| 178 |
> |
3a_{21}^e\; x & +\; 3a_{21}^o\; y & +\; 2a_{20}\; z |
| 179 |
> |
\end{array} \right), |
| 180 |
> |
\label{eq:MFE} |
| 181 |
> |
\end{equation} |
| 182 |
> |
while the gradient of the electric field in this form is: |
| 183 |
|
\begin{equation} \label{eq:grad_e2} |
| 184 |
|
\mathsf{G} = |
| 185 |
|
\begin{pmatrix} |
| 188 |
|
3a_{21}^e & 3a_{21}^o & 2a_{20} \\ |
| 189 |
|
\end{pmatrix} \\ |
| 190 |
|
\end{equation} |
| 191 |
< |
which can be factored as |
| 191 |
> |
which is also uniform over the entire space. This form for the |
| 192 |
> |
gradient can be factored as |
| 193 |
|
\begin{gather} |
| 194 |
|
\begin{aligned} |
| 195 |
|
\mathsf{G} = a_{20} |
| 226 |
|
\label{eq:intro_tensors} |
| 227 |
|
\end{gather} |
| 228 |
|
The five matrices in the expression above represent five different |
| 229 |
< |
symmetric traceless tensors of rank 2. The trace corresponds to |
| 169 |
< |
$\nabla \cdot \mathbf{E} = 0$, consistent with being in a charge-free |
| 170 |
< |
region. Using the Cartesian notation of |
| 171 |
< |
Eq. (\ref{eq:appliedPotential}), this tensor is written: |
| 172 |
< |
\begin{equation} |
| 173 |
< |
\mathsf{G} =\nabla\bf{E} |
| 174 |
< |
= \frac{g_o}{2}\left(\begin{array}{ccc} |
| 175 |
< |
2(a_1\; b_1 - \frac{cos\psi}{3}) & (a_1\; b_2 \;+ a_2\; b_1) & (a_1\; b_3 \;+ a_3\; b_1) \\ |
| 176 |
< |
(a_2\; b_1 \;+ a_1\; b_2) & 2(a_2\; b_2 \;- \frac{cos\psi}{3}) & (a_2\; b_3 \;+ a_3\; b_3) \\ |
| 177 |
< |
(a_3\; b_1 \;+ a_3\; b_2) & (a_3\; b_2 \;+ a_2\; b_3) & 2(a_3\; b_3 \;- \frac{cos\psi}{3}) |
| 178 |
< |
\end{array} \right). |
| 179 |
< |
\label{eq:GC} |
| 180 |
< |
\end{equation} |
| 229 |
> |
symmetric traceless tensors of rank 2. |
| 230 |
|
|
| 231 |
< |
It is useful to find vectors $\mathbf a$ and $\mathbf b$ that generate |
| 232 |
< |
the five types of tensors shown in Eq. (\ref{eq:intro_tensors}). If |
| 233 |
< |
the two vectors are co-linear, e.g., $\psi=0$, $\mathbf{a}=(0,0,1)$ and |
| 234 |
< |
$\mathbf{b}=(0,0,1)$, then |
| 231 |
> |
It is useful to find the Cartesian vectors $\mathbf a$ and $\mathbf b$ |
| 232 |
> |
that generate the five types of tensors shown in |
| 233 |
> |
Eq. (\ref{eq:intro_tensors}). If the two vectors are co-linear, e.g., |
| 234 |
> |
$\psi=0$, $\mathbf{a}=(0,0,1)$ and $\mathbf{b}=(0,0,1)$, then |
| 235 |
|
\begin{equation*} |
| 236 |
|
\mathsf{G} = \frac{g_0}{3} |
| 237 |
|
\begin{pmatrix} |
| 265 |
|
\end{equation*} |
| 266 |
|
The pattern is straightforward to continue for the other symmetries. |
| 267 |
|
|
| 219 |
– |
Using Eq. (\ref{eq:quad_phi}) the electric field is written: |
| 220 |
– |
\begin{equation} |
| 221 |
– |
\mathbf{E} |
| 222 |
– |
= \left(\begin{array}{ccc} |
| 223 |
– |
\left(-a_{20} + 6a_{22}^o \right) x + 6a_{22}^e y + 3a_{21}^e z \\ |
| 224 |
– |
6a_{22}^e x+(-a_{20} - 6a_{22}^o) y + 3a_{21}^e z \\ |
| 225 |
– |
3a_{21}^e x +3a_{21}^o y + 2a_{20} z |
| 226 |
– |
\end{array} \right). |
| 227 |
– |
\label{eq:MFE} |
| 228 |
– |
\end{equation} |
| 229 |
– |
while using Eq. (\ref{eq:appliedPotential}), we find: |
| 230 |
– |
\begin{equation} |
| 231 |
– |
\mathbf{E} |
| 232 |
– |
=\frac{g_o}{2} \left(\begin{array}{ccc} |
| 233 |
– |
2(a_1\; b_1 - \frac{cos\psi}{3})\;x \;+ (a_1\; b_2 \;+ a_2\; b_1)\;y + (a_1\; b_3 \;+ a_3\; b_1)\;z \\ |
| 234 |
– |
(a_2\; b_1 \;+ a_1\; b_2)\;x + 2(a_2\; b_2 \;- \frac{cos\psi}{3})\;y + (a_2\; b_3 \;+ a_3\; b_3)\;z \\ |
| 235 |
– |
(a_3\; b_1 \;+ a_3\; b_2)\;x + (a_3\; b_2 \;+ a_2\; b_3)\;y + 2(a_3\; b_3 \;- \frac{cos\psi}{3})\;z |
| 236 |
– |
\end{array} \right). |
| 237 |
– |
\label{eq:CE} |
| 238 |
– |
\end{equation} |
| 268 |
|
We find the notation of Ref. \onlinecite{Torres-del-Castillo:2006uo} |
| 269 |
< |
to be helpful when creating specific types of constant gradient |
| 270 |
< |
electric fields in simulations. For this reason, |
| 269 |
> |
helpful when creating specific types of constant gradient electric |
| 270 |
> |
fields in simulations. For this reason, |
| 271 |
|
Eqs. (\ref{eq:appliedPotential}), (\ref{eq:GC}), and (\ref{eq:CE}) are |
| 272 |
< |
used in our code. |
| 272 |
> |
implemented in our code. In the simulations using constant applied |
| 273 |
> |
gradients that are mentioned in the main text, we utilized a field |
| 274 |
> |
with the $a_{22}^e$ symmetry using vectors, $\mathbf{a}= (1, 0, 0)$ |
| 275 |
> |
and $\mathbf{b} = (0,1,0)$. |
| 276 |
|
|
| 277 |
|
\section{Point-multipolar interactions with a spatially-varying electric field} |
| 278 |
|
|
| 291 |
|
\end{align} |
| 292 |
|
where $\mathbf{r}_k$ is the local coordinate system for the object |
| 293 |
|
(usually the center of mass of object $a$). Components of vectors and |
| 294 |
< |
tensors are given using Green indices, using the Einstein repeated |
| 295 |
< |
summation notation. Note that the definition of the primitive |
| 296 |
< |
quadrupole here differs from the standard traceless form, and contains |
| 297 |
< |
an additional Taylor-series based factor of $1/2$. In Ref. |
| 298 |
< |
\onlinecite{PaperI}, we derived the forces and torques each object |
| 299 |
< |
exerts on the other objects in the system. |
| 294 |
> |
tensors are given using the Einstein repeated summation notation. Note |
| 295 |
> |
that the definition of the primitive quadrupole here differs from the |
| 296 |
> |
standard traceless form, and contains an additional Taylor-series |
| 297 |
> |
based factor of $1/2$. In Ref. \onlinecite{PaperI}, we derived the |
| 298 |
> |
forces and torques each object exerts on the other objects in the |
| 299 |
> |
system. |
| 300 |
|
|
| 301 |
|
Here we must also consider an external electric field that varies in |
| 302 |
|
space: $\mathbf E(\mathbf r)$. Each of the local charges $q_k$ in |
| 309 |
|
+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma|_{\mathbf{r}_k = 0} r_{k \delta} |
| 310 |
|
r_{k \varepsilon} + ... |
| 311 |
|
\end{equation} |
| 312 |
< |
Note that once one shrinks object $a$ to point size, the ${E}_\gamma$ |
| 313 |
< |
terms are all evaluated at the center of the object (now a |
| 314 |
< |
point). Thus later the ${E}_\gamma$ terms can be written using the |
| 315 |
< |
same global origin for all objects $a, b, c, ...$ in the system. The |
| 316 |
< |
force exerted on object $a$ by the electric field is given by, |
| 285 |
< |
|
| 312 |
> |
Note that if one shrinks object $a$ to a single point, the |
| 313 |
> |
${E}_\gamma$ terms are all evaluated at the center of the object (now |
| 314 |
> |
a point). Thus later the ${E}_\gamma$ terms can be written using the |
| 315 |
> |
same (molecular) origin for all point charges in the object. The force |
| 316 |
> |
exerted on object $a$ by the electric field is given by, |
| 317 |
|
\begin{align} |
| 318 |
|
F^a_\gamma = \sum_{k \textrm{~in~} a} E_\gamma(\mathbf{r}_k) &= \sum_{k \textrm{~in~} a} q_k \lbrace E_\gamma + \nabla_\delta E_\gamma r_{k \delta} |
| 319 |
|
+ \frac {1}{2} \nabla_\delta \nabla_\varepsilon E_\gamma r_{k \delta} |
| 322 |
|
+ Q_{a \delta \varepsilon} \nabla_\delta \nabla_\varepsilon E_\gamma + |
| 323 |
|
... |
| 324 |
|
\end{align} |
| 325 |
< |
Thus in terms of the global origin $\mathbf{r}$, ${F}_\gamma(\mathbf{r}) = C {E}_\gamma(\mathbf{r})$ etc. |
| 325 |
> |
Thus in terms of the global origin $\mathbf{r}$, ${F}_\gamma(\mathbf{r}) = C {E}_\gamma(\mathbf{r})$ etc. |
| 326 |
|
|
| 327 |
|
Similarly, the torque exerted by the field on $a$ can be expressed as |
| 328 |
|
\begin{align} |
| 354 |
|
\begin{equation} |
| 355 |
|
U(\mathbf{r}) = \mathrm{C} \phi(\mathbf{r}) - \mathrm{D}_\alpha \mathrm{E}_\alpha - \mathrm{Q}_{\alpha\beta}\nabla_\alpha \mathrm{E}_\beta + ... |
| 356 |
|
\end{equation} |
| 357 |
< |
The results has been summarized in Table I. |
| 357 |
> |
These results have been summarized in Table \ref{tab:UFT}. |
| 358 |
|
|
| 359 |
|
\begin{table} |
| 360 |
|
\caption{Potential energy $(U)$, force $(\mathbf{F})$, and torque |
| 361 |
< |
$(\mathbf{\tau})$ expressions for a multipolar site $\mathrm{r}$ in an |
| 362 |
< |
electric field, $\mathbf{E}(\mathbf{r})$. |
| 363 |
< |
\label{tab:UFT}} |
| 364 |
< |
\begin{tabular}{r|ccc} |
| 361 |
> |
$(\mathbf{\tau})$ expressions for a multipolar site at $\mathbf{r}$ in an |
| 362 |
> |
electric field, $\mathbf{E}(\mathbf{r})$ using the definitions of the multipoles in Eqs. (\ref{eq:charge}), (\ref{eq:dipole}) and (\ref{eq:quadrupole}). |
| 363 |
> |
\label{tab:UFT}} |
| 364 |
> |
\begin{tabular}{r|C{3cm}C{3cm}C{3cm}} |
| 365 |
|
& Charge & Dipole & Quadrupole \\ \hline |
| 366 |
|
$U(\mathbf{r})$ & $C \phi(\mathbf{r})$ & $-\mathbf{D} \cdot \mathbf{E}(\mathbf{r})$ & $- \mathsf{Q}:\nabla \mathbf{E}(\mathbf{r})$ \\ |
| 367 |
|
$\mathbf{F}(\mathbf{r})$ & $C \mathbf{E}(\mathbf{r})$ & $\mathbf{D} \cdot \nabla \mathbf{E}(\mathbf{r})$ & $\mathsf{Q} : \nabla\nabla\mathbf{E}(\mathbf{r})$ \\ |
| 369 |
|
\end{tabular} |
| 370 |
|
\end{table} |
| 371 |
|
|
| 341 |
– |
|
| 372 |
|
\section{Boltzmann averages for orientational polarization} |
| 373 |
< |
The dielectric properties of the system is mainly arise from two |
| 373 |
> |
The dielectric properties of the system mainly arise from two |
| 374 |
|
different ways: i) the applied field distort the charge distributions |
| 375 |
|
so it produces an induced multipolar moment in each molecule; and ii) |
| 376 |
|
the applied field tends to line up originally randomly oriented |
| 442 |
|
\braket{q_{\alpha\beta}} = \frac{ \int d\Omega \left(1 + \frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu }{k_B T}\right)q_{\alpha\beta}}{ \int d\Omega \left(1 + \frac{\eta_{\mu\mu'}\;\eta_{\nu\nu'}\;{q}^*_{\mu'\nu'}\;\partial_\nu E_\mu }{k_B T}\right)}, |
| 443 |
|
\end{equation} |
| 444 |
|
where $\eta_{\alpha\alpha'}$ is the inverse of the rotation matrix that transforms |
| 445 |
< |
the body fixed co-ordinates to the space co-ordinates, |
| 446 |
< |
\[\eta_{\alpha\alpha'} |
| 447 |
< |
= \left(\begin{array}{ccc} |
| 448 |
< |
cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & -cos\theta\; cos\psi\; sin\phi - cos\phi\; sin\psi & sin\theta\; sin\phi \\ |
| 449 |
< |
cos\psi\; sin\phi + cos\theta\; cos\phi \; sin\psi & cos\theta\; cos\phi\; cos\psi - sin\phi\; sin\psi & -cos\phi\; sin\theta \\ |
| 450 |
< |
sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta |
| 451 |
< |
\end{array} \right).\] |
| 445 |
> |
the body fixed co-ordinates to the space co-ordinates. |
| 446 |
> |
% \[\eta_{\alpha\alpha'} |
| 447 |
> |
% = \left(\begin{array}{ccc} |
| 448 |
> |
% cos\phi\; cos\psi - cos\theta\; sin\phi\; sin\psi & -cos\theta\; cos\psi\; sin\phi - cos\phi\; sin\psi & sin\theta\; sin\phi \\ |
| 449 |
> |
% cos\psi\; sin\phi + cos\theta\; cos\phi \; sin\psi & cos\theta\; cos\phi\; cos\psi - sin\phi\; sin\psi & -cos\phi\; sin\theta \\ |
| 450 |
> |
% sin\theta\; sin\psi & -cos\psi\; sin\theta & cos\theta |
| 451 |
> |
% \end{array} \right).\] |
| 452 |
> |
|
| 453 |
|
Integration of 1st and 2nd terms in the denominator gives $8 \pi^2$ |
| 454 |
|
and $8 \pi^2 /3\;{\nabla}.\mathbf{E}\; Tr(q^*) $ respectively. The |
| 455 |
|
second term vanishes for charge free space, ${\nabla}.\mathbf{E} \; = \; 0$. Similarly integration of the |
| 468 |
|
where $ \alpha_q = \frac{{\bar{q_o}}^2}{15k_BT} $ is a molecular quadrupole polarizablity and ${\bar{q_o}}^2= |
| 469 |
|
3{q}^*_{\alpha'\beta'}{q}^*_{\beta'\alpha'}-{q}^*_{\alpha'\alpha'}{q}^*_{\beta'\beta'}$ is a square of the net quadrupole moment of a molecule. |
| 470 |
|
|
| 440 |
– |
% \section{External application of a uniform field gradient} |
| 441 |
– |
% \label{Ap:fieldOrGradient} |
| 442 |
– |
|
| 443 |
– |
% To satisfy the condition $ \nabla \cdot \mathbf{E} = 0 $, within the box of molecules we have taken electrostatic potential in the following form |
| 444 |
– |
% \begin{equation} |
| 445 |
– |
% \begin{split} |
| 446 |
– |
% \phi(x, y, z) =\; &-g_o \left(\frac{1}{2}(a_1\;b_1 - \frac{cos\psi}{3})\;x^2+\frac{1}{2}(a_2\;b_2 - \frac{cos\psi}{3})\;y^2 + \frac{1}{2}(a_3\;b_3 - \frac{cos\psi}{3})\;z^2 \right. \\ |
| 447 |
– |
% & \left. + \frac{(a_1\;b_2 + a_2\;b_1)}{2} x\;y + \frac{(a_1\;b_3 + a_3\;b_1)}{2} x\;z + \frac{(a_2\;b_3 + a_3\;b_2)}{2} y\;z \right), |
| 448 |
– |
% \end{split} |
| 449 |
– |
% \label{eq:appliedPotential} |
| 450 |
– |
% \end{equation} |
| 451 |
– |
% where $a = (a_1, a_2, a_3)$ and $b = (b_1, b_2, b_3)$ are basis vectors determine coefficients in x, y, and z direction. And $g_o$ and $\psi$ are overall strength of the potential and angle between basis vectors respectively. The electric field derived from the above potential is, |
| 452 |
– |
% \[\mathbf{E} |
| 453 |
– |
% = \frac{g_o}{2} \left(\begin{array}{ccc} |
| 454 |
– |
% 2(a_1\; b_1 - \frac{cos\psi}{3})\;x \;+ (a_1\; b_2 \;+ a_2\; b_1)\;y + (a_1\; b_3 \;+ a_3\; b_1)\;z \\ |
| 455 |
– |
% (a_2\; b_1 \;+ a_1\; b_2)\;x + 2(a_2\; b_2 \;- \frac{cos\psi}{3})\;y + (a_2\; b_3 \;+ a_3\; b_2)\;z \\ |
| 456 |
– |
% (a_3\; b_1 \;+ a_3\; b_2)\;x + (a_3\; b_2 \;+ a_2\; b_3)y + 2(a_3\; b_3 \;- \frac{cos\psi}{3})\;z |
| 457 |
– |
% \end{array} \right).\] |
| 458 |
– |
% The gradient of the applied field derived from the potential can be written in the following form, |
| 459 |
– |
% \[\nabla\mathbf{E} |
| 460 |
– |
% = \frac{g_o}{2}\left(\begin{array}{ccc} |
| 461 |
– |
% 2(a_1\; b_1 - \frac{cos\psi}{3}) & (a_1\; b_2 \;+ a_2\; b_1) & (a_1\; b_3 \;+ a_3\; b_1) \\ |
| 462 |
– |
% (a_2\; b_1 \;+ a_1\; b_2) & 2(a_2\; b_2 \;- \frac{cos\psi}{3}) & (a_2\; b_3 \;+ a_3\; b_2) \\ |
| 463 |
– |
% (a_3\; b_1 \;+ a_3\; b_2) & (a_3\; b_2 \;+ a_2\; b_3) & 2(a_3\; b_3 \;- \frac{cos\psi}{3}) |
| 464 |
– |
% \end{array} \right).\] |
| 465 |
– |
|
| 466 |
– |
|
| 467 |
– |
|
| 471 |
|
\section{Gradient of the field due to quadrupolar polarization} |
| 472 |
|
\label{singularQuad} |
| 473 |
|
In this section, we will discuss the gradient of the field produced by |
| 569 |
|
polarization produces zero net gradient of the field inside the |
| 570 |
|
sphere. |
| 571 |
|
|
| 572 |
+ |
\section{Geometric Factors for Two Embedded Point Charges} |
| 573 |
|
|
| 574 |
|
\bibliography{dielectric_new} |
| 575 |
|
\end{document} |
